Finding Bit-Based Division Property for Ciphers with Complex Linear Layers

Abstract

The bit-based division property (BDP) is the most effective technique for finding integral characteristics of symmetric ciphers. Recently, automatic search tools have become one of the most popular approaches to evaluating the security of designs against many attacks. Constraint-aided automatic tools for the BDP have been applied to many ciphers with simple linear layers like bit-permutation. Constructing models of complex linear layers accurately and efficiently remains hard. A straightforward method proposed by Sun et al. (called the S method), decomposes a complex linear layer into basic operations like COPY and XOR, then models them one by one. However, this method can easily insert invalid division trails into the solution pool, which results in a quicker loss of the balanced property than the cipher itself would. In order to solve this problem, Zhang and Rijmen propose the ZR method to link every valid trail with an invertible sub-matrix of the matrix corresponding to the linear layer, and then generate linear inequalities to represent all the invertible sub-matrices. Unfortunately, the ZR method is only applicable to invertible binary matrices (defined in Definition 3).To avoid generating a huge number of inequalities for all the sub-matrices, we build a new model that only includes that the sub-matrix corresponding to a valid trail should be invertible. The computing scale of our model can be tackled by most of SMT/SAT solvers, which makes our method practical. For applications, we improve the previous BDP for LED and MISTY1. We also give the 7-round BDP results for Camellia with FL/FL−1, which is the longest to date.Furthermore, we remove the restriction of the ZR method that the matrix has to be invertible, which provides more choices for future designs. Thanks to this, we also reproduce 5-round key-dependent integral distinguishers proposed at Crypto 2016 which cannot be obtained by either the S or ZR methods.

Massive Superpoly Recovery with Nested Monomial Predictions

“…The dominant part the complexity for the key recovery attack is the exhaustive search, which is 2 76.32 Trivium encryptions. 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,23,25,27,30,32,34,36,38,40,42,45,47,49,51,53,55, 57, 60, 62, 64, 66, 68, 70, 72, 77, 75, 79 I2 52 0, 1, 2, 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,…”

“…From then on, a series of researches on extending the application scope or increasing the accuracy of the algorithms for detecting division properties were conducted [15,14,25,24]. To capture not only balanced but also constant output bits as well as some cancellation characteristics ignored by the conventional bitbased division property, the so-called three-subset bit-based division property was proposed [40].…”

“…Then the complexity is about 2 78.6 , which is the best key recovery attack for 842-round Trivium. In this paper, we consider the following 56-dimension cube indices I = { 0, 1, 2, 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,36,38,40,42,45,47,49,51,53,55,57,60,62,64,66,68,70,72,77,75, 79 }.…”

“…Cube indices we use for the superpoly recovery of 843-round Trivium ,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 38, 40, 42, 45, 47, 49, 51, 53, 55, 57, 60, 62, 64, 66, 68, 70, 72, 77, 75, 79 I3 54 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 42, 45, 47, 49, 51, 53, 55, 57, 60, 62, 64, 66, 68, 70, 72, 77, 75, 79 I4 76 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,21,22,23,24,25,26,27,28,29,30,31,32,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 75, 77, 79 …”

“…Cube indices we use for the superpoly recovery of 840-round Trivium ,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19,21,23,25,27,30,32,34,36,38,40,42,45,47,49, 51, 53, 55, 57, 60, 62, 64, 66, 68, 70, 72, 77, 75, 79 …”